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Zeta functions have been a powerful tool in mathematics over the last two centuries. This book considers a new class of non-commutative zeta functions which encode the structure of the subgroup lattice in infinite groups. The book explores the analytic behaviour of these functions together with an investigation of functional equations. Many important examples of zeta functions are calculated and recorded providing an important data base of explicit examples and methods for calculation.

Group theory. --- Functions, Zeta. --- Rings (Algebra) --- Noncommutative algebras. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebras, Noncommutative --- Non-commutative algebras --- Algebraic rings --- Ring theory --- Algebraic fields --- Zeta functions --- Rings (Algebra). --- Number theory. --- Algebra. --- Group Theory and Generalizations. --- Number Theory. --- Non-associative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Number study --- Numbers, Theory of --- Functions, Zeta --- Group theory --- Noncommutative algebras --- 512.54 --- 512.54 Groups. Group theory --- Groups. Group theory --- Nonassociative rings.